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Exercise 7. Let X be a topological space. Suppose B(x) is a basis of neighborhoods of x = X. a) For a sequence (xn)neN in X, show that x is a limit of (xn)neN if and only if every V = B(x) contains all the xn except maybe a finite number of them. h) In nautioular hon lima if 1 no m

Exercise 7. Let X be a topological space. Suppose B(x) is a basis of neighborhoods of x = X. a) For a sequence (xn)neN in X, show that x is a limit of (xn)neN if and only if every V = B(x) contains all the xn except maybe a finite number of them. h) In nautioular hon lima if 1 no m

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.1: Polynomials Over A Ring
Problem 21E: Describe the kernel of epimorphism in Exercise 20. Consider the mapping :Z[ x ]Zk[ x ] defined by...
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Exercise 7 Need parts a b and c
Exercise 7. Let X be a topological space. Suppose B(x) is a basis of neighborhoods of x € X. a) For a sequence (n)neN in X, show that x is a limit of (xn)neN if and only if every V = B(x) contains all the xn except maybe a finite number of them. b) In particular, show that x is a limit of (n)neN if and only if every open set U containing x contains all the xn except maybe a finite number of them. c) Let (n)neN be a sequence in the metric space (X, d). Show that x = X is a limit of (n)neN if and only if for every e > 0 there exists ne EN such that d(xn, x) ≤ e, for all n ≥ ne. This is equivalent to d(xn, x) → 0 in R.
Transcribed Image Text:Exercise 7. Let X be a topological space. Suppose B(x) is a basis of neighborhoods of x € X. a) For a sequence (n)neN in X, show that x is a limit of (xn)neN if and only if every V = B(x) contains all the xn except maybe a finite number of them. b) In particular, show that x is a limit of (n)neN if and only if every open set U containing x contains all the xn except maybe a finite number of them. c) Let (n)neN be a sequence in the metric space (X, d). Show that x = X is a limit of (n)neN if and only if for every e > 0 there exists ne EN such that d(xn, x) ≤ e, for all n ≥ ne. This is equivalent to d(xn, x) → 0 in R.
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